A comparison of numerical solutions of the one-dimensional unsaturated—saturated flow and mass transport equations

We made use of groundwater flow and mass transport equations to investigate the crucial potential risk of water pollution from hydraulic fracturing especially in the case of the Karoo system in South Africa. This paper shows that the upward migration of fluids will depend on the apertures of the cement cracks and fractures in the rock formation. The greater the apertures, the quicker the movement of the fluid. We presented a novel sampling method, which is the combination of the Monte Carlo and the Latin hypercube sampling. The method was used for uncertainties analysis of the apertures in the groundwater and mass transport equations. The study reveals that, in the case of the Karoo, fracking will only be successful if and only if the upward methane and fracking fluid migration can be controlled, for example, by plugging the entire fracked reservoir with cement.

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Diffusion-Limited Cell Dehydration: Analytical and Numerical Solutions for a Planar Model

10.1115/imece1999-0600 ◽

1999 ◽

Author(s):

Alexander V. Kasharin ◽

Jens O. M. Karlsson

Keyword(s):

Analytical Solution ◽

Numerical Solutions ◽

Planar System ◽

One Dimensional ◽

Diffusion Limited ◽

Dimensional Diffusion ◽

Constant Diffusivity ◽

A Cell ◽

Cell Dehydration ◽

The One

Abstract The process of diffusion-limited cell dehydration is modeled for a planar system by writing the one-dimensional diffusion-equation for a cell with moving, semipermeable boundaries. For the simplifying case of isothermal dehydration with constant diffusivity, an approximate analytical solution is obtained by linearizing the governing partial differential equations. The general problem must be solved numerically. The Forward Time Center Space (FTCS) and Crank-Nicholson differencing schemes are implemented, and evaluated by comparison with the analytical solution. Putative stability criteria for the two algorithms are proposed based on numerical experiments, and the Crank-Nicholson method is shown to be accurate for a mesh with as few as six nodes.

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Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

Journal of Fluid Mechanics ◽

10.1017/s0022112099006084 ◽

1999 ◽

Vol 396 ◽

pp. 223-256 ◽

Cited By ~ 49

Author(s):

B. S. BROOK ◽

S. A. E. G. FALLE ◽

T. J. PEDLEY

Keyword(s):

Shallow Water ◽

Numerical Solutions ◽

Godunov Method ◽

Test Case ◽

Interesting Phenomenon ◽

One Dimensional ◽

Collapsible Tubes ◽

Highly Nonlinear ◽

Pressure Area ◽

The One

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.

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FULLY DISCRETE SCHEMES FOR THE SCHRÖDINGER EQUATION: DISPERSIVE PROPERTIES

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202507002029 ◽

2007 ◽

Vol 17 (04) ◽

pp. 567-591 ◽

Cited By ~ 6

Author(s):

LIVIU I. IGNAT

Keyword(s):

Schrödinger Equation ◽

Schrodinger Equation ◽

Numerical Solutions ◽

Continuous Model ◽

Local Smoothing ◽

Euler Approximation ◽

One Dimensional ◽

Fully Discrete ◽

Backward Euler ◽

The One

We consider fully discrete schemes for the one-dimensional linear Schrödinger equation and analyze whether the classical dispersive properties of the continuous model are presented in these approximations. In particular, Strichartz estimates and the local smoothing of the numerical solutions are analyzed. Using a backward Euler approximation of the linear semigroup we introduce a convergent scheme for the nonlinear Schrödinger equation with nonlinearities which cannot be treated by energy methods.

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Genuinely Multi-Dimensional Non-Dissipative Finite-Volume Schemes for Transport

International Journal of Applied Mathematics and Computer Science ◽

10.2478/v10006-007-0026-z ◽

2007 ◽

Vol 17 (3) ◽

pp. 321-328 ◽

Cited By ~ 1

Author(s):

Bruno Després ◽

Frédéric Lagoutière

Keyword(s):

Finite Volume ◽

Transverse Direction ◽

Transport Equations ◽

Discrete Solution ◽

Finite Volume Schemes ◽

Time Step ◽

One Dimensional ◽

Transport Velocity ◽

Volume Algorithm ◽

The One

Genuinely Multi-Dimensional Non-Dissipative Finite-Volume Schemes for TransportWe develop a new multidimensional finite-volume algorithm for transport equations. This algorithm is both stable and non-dissipative. It is based on a reconstruction of the discrete solution inside each cell at every time step. The proposed reconstruction, which is genuinely multidimensional, allows recovering sharp profiles in both the direction of the transport velocity and the transverse direction. It constitutes an extension of the one-dimensional reconstructions analyzed in (Lagoutière, 2005; Lagoutière, 2006).

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